University:
York University
Course:
MATH 1013 | Applied Calculus I
Academic year:

2022
Views:

323

Pages:

8
Author:

Nick T.

Mock Midterm 1 Real Midterm 1 (-∞, 0) U (2, ∞) U(2, ∞) X f(x) = x^3 - 2x^2 - x + 2 -x^3 + 2x^2 + x - 2 x^2(x-2) - 1(x-2) (x^2-1)(x-2) (x+1)(x-1)(x-2) x-intercepts: -1, 1, 2 domain: (-∞, ∞) If point (1,2) is on the graph of an even function, what other point must also be on the graph? (-1, 2) (even function, y-axis reflection) (1, 2) odd function (-1, -2) (x-axis reflection) Multiple choice: 1 chance/ Numbers only: 2 chances/ Equations: 3 chances Shifting a Graph f(x-3): 3 units to the left f(3x): vertical stretch x3 f(x/3): horizontal shrink -f(x): reflection x-axis f(-x): reflection y-axis f(x) + 1: vertical shift up 1 f(x) - 1: vertical shift down 1 10^x (add 2 inside) 2 units left Inverse y = x^2 + 5 x = y^2 + 5 x - 5 = y^2 y = √(x-5) The answer cannot be negative. 7. One-to-one functions 8. Log(5) Ln(5) = Log(5)^Ln(5) Log 5^Ln(5) = x 5^x = 5^(Ln(5)) x = Ln(5) 9. Laws of logs Study Ln(a*b) = Ln(a) + Ln(b) Ln(a/b) = Ln(a) - Ln(b) Ln(a^b) = b Ln(a) 10. Limits lim(x->-24) (√(x^2+49) - 7) / (x+24) Rationalize the numerator: lim(x->-24) ((√(x^2+49) - 7) / (x+24)) * ((√(x^2+49) + 7) / (√(x^2+49) + 7))) lim(x->-24) (x^2+49-49) / ((x+24)(√(x^2+49) + 7)) lim(x->-24) x / ((x+24)(√(x^2+49) + 7)) -24 / (0 + 14) = -12/7 11. Limit lim(x->4) (x^2-16) / (x-4) lim(x->4) (x+4)(x-4) / (x-4) lim(x->4) (x+4) = 8 Study this!! Limits: lim(x->∞) (x^2 + x) / (x^2 + e^x) = 0 lim(x->0) arctan(x) / x = 0 Graph of arctan(x): arctan(x) graph Function Analysis: y = (9+x^4) / (x^2 - x^4) Vertical Asymptotes: x = -1, 1 Horizontal Asymptote: y = -1 lim(x->∞) (9+x^4) / (x^2 - x^4) = -1 Finding x-intercepts: x^2 - x^4 = 0 x^2(1 - x^2) = 0 x = 0, ±1